AI 中文总结
研究具有Burgers型非线性的耗散偏微分方程主从同步的随机稳定性，通过傅里叶截断等方法，先证确定性同步流形局部指数稳定性，再引入噪声分析偏差，得到均方界等结果，还将其用于数据同化并与滤波器比较。
AI 中文摘要
我们研究了一类具有Burgers型对流非线性和多项式线性微分算子的非线性耗散演化方程的主从同步的随机稳定性。该方程族包括Burgers、Kuramoto-Sivashinsky、Kawahara、Benney-Lin和Nikolaevskiy方程。在周期边界条件下，通过有限维傅里叶截断表示每个方程，得到一个与由主系统观测驱动的从系统耦合的复状态向量。首先在耦合强度的简单条件下建立确定性同步流形的局部指数稳定性。然后在耦合信号中引入观测噪声，将从系统转化为伊藤扩散并防止精确同步。分析集中在随机同步误差与指数稳定的确定性参考误差之间的偏差Δ(t)。证明了关于||Δ(t)||²的O(σ²)有限时间均方界以及相应的尾概率估计。在全局单侧耗散假设下，去除局部化并获得O(σ²)时间一致界。这些估计在傅里叶空间中导出并通过Parseval关系转换到物理空间。最后，将随机从系统动力学解释为基于同步的数据同化方案，并将其结构与集合卡尔曼-布西滤波器进行比较，强调规定的稳定性导向增益和基于自适应协方差增益之间的差异。
英文摘要
We investigate the stochastic stability of master--slave synchronization for a class of nonlinear dissipative PDEs with Burgers-type nonlinearity and polynomial linear operator, including the Burgers, Kuramoto--Sivashinsky, Kawahara, Benney--Lin, and Nikolaevskiy equations. Under periodic boundary conditions, each equation is represented by a finite-dimensional Fourier truncation coupled to a slave driven by observed master data. We establish local exponential stability of the deterministic zero-error synchronization manifold under a simple coupling condition. Introducing observational noise in the coupling transforms the slave into an Itô diffusion, preventing exact synchronization. We analyze the deviation $Δ(t)$ between the stochastic error and the exponentially stable deterministic reference error, proving an $\mathcal{O}(σ^2)$ finite-time mean-square bound localized near the synchronization manifold, with a tail-probability estimate. Under global one-sided dissipativity, the localization is removed and a time-uniform $\mathcal{O}(σ^2)$ bound is obtained. Bounds are derived in Fourier space and transferred to the physical domain via Parseval's identity. Under uniform Sobolev and Galerkin stability assumptions, we also bound the error with respect to the infinite-dimensional master uniformly in truncation order. Results are illustrated by numerical simulations of Kawahara and Nikolaevskiy master--slave systems.
